On the Petersson scalar product of arbitrary modular forms

We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight \(k\ge 2\) for a finite index subgroup of the modular group. We show that Hecke operators have the same adjoints with respect to this inner product as for cusp forms, and we show t...

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Bibliographic Details
Published in:arXiv.org 2013-11
Main Authors: Pasol, Vicentiu, Popa, Alexandru A
Format: Article
Language:English
Subjects:
Adjoints
Analytic functions
Operators (mathematics)
Subgroups
Weight
Online Access:Get full text
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Summary:We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight \(k\ge 2\) for a finite index subgroup of the modular group. We show that Hecke operators have the same adjoints with respect to this inner product as for cusp forms, and we show that the Petersson product is nondegenerate for \(\Gamma_1(N)\) and \(k>2\). For \(k=2\) we give examples when it is degenerate, and when it is nondegenerate.
ISSN:2331-8422